Contact integral geometry and the Heisenberg algebra

Abstract

Generalizing Weyl's tube formula and building on Chern's work, Alesker reinterpreted the Lipschitz-Killing curvature integrals as a family of valuations (finitely-additive measures with good analytic properties), attached canonically to any Riemannian manifold, which is universal with respect to isometric embeddings. In this note, we uncover a similar structure for contact manifolds. Namely, we show that a contact manifold admits a canonical family of generalized valuations, which are universal under contact embeddings. Those valuations assign numerical invariants to even-dimensional submanifolds, which in a certain sense measure the curvature at points of tangency to the contact structure. Moreover, these valuations generalize to the class of manifolds equipped with the structure of a Heisenberg algebra on their cotangent bundle. Pursuing the analogy with Euclidean integral geometry, we construct symplectic-invariant distributions on Grassmannians to produce Crofton formulas on the contact sphere. Using closely related distributions, we obtain Crofton formulas also in the linear symplectic space.

Uniqueness of Curvature Measures in Pseudo-Riemannian Geometry

“…Important examples of Weyl functors are the intrinsic volumes of Riemannian manifolds, taking values in smooth valuations, and the Lipschitz-Killing curvature measures [22,25]. For a different example, a family of Weyl functors on contact manifolds with values in generalized valuations was described in [21].…”

Curvature Measures of Pseudo-Riemannian Manifolds

“…The product satisfies a version of Poincaré duality, which gives rise to the notion of generalized valuations on a manifold. Generalized valuations appear quite naturally in Hadwiger-type theorems for noncompact groups such as the Lorentz group [10,17,29].…”

“…Some conjectures on the behaviour of Lipschitz-Killing curvatures under Gromov-Hausdorff convergence with potential applications to the theory of Alexandrov spaces with curvature bounded from below are contained in the recent paper [7]. Let us also mention [29], where analogues of the Lipschitz-Killing curvature measures that are natural under embeddings have been constructed for contact manifolds.…”

The Weyl principle on the Finsler frontier

“…A guiding idea in valuation theory, usually referred to as the Weyl principle, has recently emerged: when restricted to a subspace X, valuations on an ambient space M may often be reconstructed from the basic geometry of X induced by the immersion into M . The prototypical application is Weyl's theorem described above, but more recently several other instances have surfaced: immersions of contact and dual Heisenberg manifolds and restriction of their canonical valuations [18]; restriction of the invariant valuations on complex space forms to totally real submanifolds [15,Lemma 4.4]; and immersions of pseudo-Riemannian manifolds and the restriction of their canonical valuations [14].…”